254 XIV INTEGRATION IN LOCALLY COMPACT GROUPS 1,^2)^1(^1) f r = d/22(x2) J J = f(xl9 (*i> x2) ^i(X by virtue of (13.21.2). Hence the result. In particular, Lebesgue measure on R" (13.21.19) is a Haar measure on the additive group R". PROBLEMS 1. Let G be a locally compact group, p, a left Haar measure on G, A a subset of G, and B a relatively compact /z-integrable subset of G such that ^u(B) > 0. Show that, if ft*(AB) < -f oo, then A is relatively compact (imitate the proof of (14.2.3)). 2. (a) Let G be a discrete subgroup of rank n in the additive group R", acting on R" by translations. Show that, if A is Lebesgue measure, the number A(G) (Section 14.1, Problem 6(b)) is equal to the absolute value of the determinant (with respect to the canonical basis of R") of a Z-basis of G (use Problem 6(a) of Section 14.1). Deduce that if A is a closed symmetric convex set in R" with nonempty interior (Section 12.14, Problem 11) such that A(A) ^ 2"A(G), then A n G contains a point other than 0 (Minkowski's theorem). n (b) Let Ui : (jc/)f— » ^ CijXj (1 £ i ^ m) be linear forms on RM with integer coefficients j=i €„, and suppose that m < n. Let p be an integer > 1, and let A be a symmetric convex set in R" with nonempty interior. Show that, for each r > 0 satisfying A(A)r" ^ 2npm, there exists a point x ^ 0 in rA with integer coefficients, such that ut(x) = 0 (mod p) for 1 <£ / < m. (Apply Minkowski's theorem to the subgroup GO of Z" consisting of all z e Z" such that ut(z) ^ 0 (mod p) for 1 <; / g w, and use Problem 6(d) of Section 1 4.1 .) In particular, show that if c1? c2 are any two integers, there exist integers xt, x2, not both zero, such that |*i| ^Vp, \x2\ ^^/p and c^j. -f c2x2 =0 (mod p) (Thue's theorem). (c) Let a, b be two integers. Use (b) to show that there exist integers xit x2 , #3, ^4, not all zero, such that 55 x$ (mod />), bxi — ax2 « x4 (mod p) and y = xl -f xl + xl -f- xl Show that if p is prime one can find two integers a, b such that a2 + b2 -f 1 = 0 (mod/?) (assuming/? is odd, observe that when z takes the J(P+ 1) integer valuesces /*, on sV for all s e G. Clearly a is left-